Problem about in integral domain?

Question

Let R denote the field of real numbers and let A=R[x,y,z]/(x^2+y^2+z^2-1)
1> show A is an integral domain?
2> Prove X-1 +(x^2+y^2+z^2-1) is a prime element in A
3> Examine if A is a PID

Answer 1

tell me 1 and 1 = ?.

Answer 2

1. it is sufficient to show that P = x^2+y^2+z^2-1 is prime element in R[x,y,z]. So:
x^2+y^2+z^2-1 = fg, examine the coefficients.....
you obtain that f or g are equivalent with P.

2. the same X-1 = fg.......

3. Examine the ideal (x,y). Is it principal?
If it is then x = f q_1 + Ps.
y = f q _2 + Pt.
Now examine the degrees of the elements that appear in Ps, Pt, f q_1 ,f q _2 .

Answer 3

1. it is sufficient to show that P = x^2+y^2+z^2-1 is prime element in R[x,y,z]. So:
x^2+y^2+z^2-1 = fg, examine the coefficients.....
you obtain that f or g are equivalent with P.

2. the same X-1 = fg.......

3. Examine the ideal (x,y). Is it principal?
If it is then x = f q_1 + Ps.
y = f q _2 + Pt.
Now examine the degrees of the elements that appear in Ps, Pt, f q_1 ,f q _2 .

My advice: it is not principal